This paper investigates Frequentist consistency properties of the posterior distributions constructed via Generalized Variational Inference (GVI). A number of generic and novel strategies are given for proving consistency, relying on the theory of $\Gamma$-convergence. Specifically, this paper shows that under minimal regularity conditions, the sequence of GVI posteriors is consistent and collapses to a point mass at the population-optimal parameter value as the number of observations goes to infinity. The results extend to the latent variable case without additional assumptions and hold under misspecification. Lastly, the paper explains how to apply the results to a selection of GVI posteriors with especially popular variational families. For example, consistency is established for GVI methods using the mean field normal variational family, normal mixtures, Gaussian process variational families as well as neural networks indexing a normal (mixture) distribution.
In recent years, the field of machine learning has made phenomenal progress in the pursuit of simulating real-world data generation processes. One notable example of such success is the variational autoencoder (VAE). In this work, with a small shift in perspective, we leverage and adapt VAEs for a different purpose: uncertainty quantification in scientific inverse problems. We introduce UQ-VAE: a flexible, adaptive, hybrid data/model-informed framework for training neural networks capable of rapid modelling of the posterior distribution representing the unknown parameter of interest. Specifically, from divergence-based variational inference, our framework is derived such that most of the information usually present in scientific inverse problems is fully utilized in the training procedure. Additionally, this framework includes an adjustable hyperparameter that allows selection of the notion of distance between the posterior model and the target distribution. This introduces more flexibility in controlling how optimization directs the learning of the posterior model. Further, this framework possesses an inherent adaptive optimization property that emerges through the learning of the posterior uncertainty.
This paper presents a variational Bayesian kernel selection (VBKS) algorithm for sparse Gaussian process regression (SGPR) models. In contrast to existing GP kernel selection algorithms that aim to select only one kernel with the highest model evidence, our proposed VBKS algorithm considers the kernel as a random variable and learns its belief from data such that the uncertainty of the kernel can be interpreted and exploited to avoid overconfident GP predictions. To achieve this, we represent the probabilistic kernel as an additional variational variable in a variational inference (VI) framework for SGPR models where its posterior belief is learned together with that of the other variational variables (i.e., inducing variables and kernel hyperparameters). In particular, we transform the discrete kernel belief into a continuous parametric distribution via reparameterization in order to apply VI. Though it is computationally challenging to jointly optimize a large number of hyperparameters due to many kernels being evaluated simultaneously by our VBKS algorithm, we show that the variational lower bound of the log-marginal likelihood can be decomposed into an additive form such that each additive term depends only on a disjoint subset of the variational variables and can thus be optimized independently. Stochastic optimization is then used to maximize the variational lower bound by iteratively improving the variational approximation of the exact posterior belief via stochastic gradient ascent, which incurs constant time per iteration and hence scales to big data. We empirically evaluate the performance of our VBKS algorithm on synthetic and massive real-world datasets.
We place an Indian Buffet process (IBP) prior over the structure of a Bayesian Neural Network (BNN), thus allowing the complexity of the BNN to increase and decrease automatically. We further extend this model such that the prior on the structure of each hidden layer is shared globally across all layers, using a Hierarchical-IBP (H-IBP). We apply this model to the problem of resource allocation in Continual Learning (CL) where new tasks occur and the network requires extra resources. Our model uses online variational inference with reparameterisation of the Bernoulli and Beta distributions, which constitute the IBP and H-IBP priors. As we automatically learn the number of weights in each layer of the BNN, overfitting and underfitting problems are largely overcome. We show empirically that our approach offers a competitive edge over existing methods in CL.
We propose a scalable Bayesian preference learning method for jointly predicting the preferences of individuals as well as the consensus of a crowd from pairwise labels. Peoples' opinions often differ greatly, making it difficult to predict their preferences from small amounts of personal data. Individual biases also make it harder to infer the consensus of a crowd when there are few labels per item. We address these challenges by combining matrix factorisation with Gaussian processes, using a Bayesian approach to account for uncertainty arising from noisy and sparse data. Our method exploits input features, such as text embeddings and user metadata, to predict preferences for new items and users that are not in the training set. As previous solutions based on Gaussian processes do not scale to large numbers of users, items or pairwise labels, we propose a stochastic variational inference approach that limits computational and memory costs. Our experiments on a recommendation task show that our method is competitive with previous approaches despite our scalable inference approximation. We demonstrate the method's scalability on a natural language processing task with thousands of users and items, and show improvements over the state of the art on this task. We make our software publicly available for future work.
Data poisoning attacks compromise the integrity of machine-learning models by introducing malicious training samples to influence the results during test time. In this work, we investigate backdoor data poisoning attack on deep neural networks (DNNs) by inserting a backdoor pattern in the training images. The resulting attack will misclassify poisoned test samples while maintaining high accuracies for the clean test-set. We present two approaches for detection of such poisoned samples by quantifying the uncertainty estimates associated with the trained models. In the first approach, we model the outputs of the various layers (deep features) with parametric probability distributions learnt from the clean held-out dataset. At inference, the likelihoods of deep features w.r.t these distributions are calculated to derive uncertainty estimates. In the second approach, we use Bayesian deep neural networks trained with mean-field variational inference to estimate model uncertainty associated with the predictions. The uncertainty estimates from these methods are used to discriminate clean from the poisoned samples.
Stochastic variational inference (SVI) plays a key role in Bayesian deep learning. Recently various divergences have been proposed to design the surrogate loss for variational inference. We present a simple upper bound of the evidence as the surrogate loss. This evidence upper bound (EUBO) equals to the log marginal likelihood plus the KL-divergence between the posterior and the proposal. We show that the proposed EUBO is tighter than previous upper bounds introduced by $\chi$-divergence or $\alpha$-divergence. To facilitate scalable inference, we present the numerical approximation of the gradient of the EUBO and apply the SGD algorithm to optimize the variational parameters iteratively. Simulation study with Bayesian logistic regression shows that the upper and lower bounds well sandwich the evidence and the proposed upper bound is favorably tight. For Bayesian neural network, the proposed EUBO-VI algorithm outperforms state-of-the-art results for various examples.
Variational inference offers scalable and flexible tools to tackle intractable Bayesian inference of modern statistical models like Bayesian neural networks and Gaussian processes. For largely over-parameterized models, however, the over-regularization property of the variational objective makes the application of variational inference challenging. Inspired by the literature on kernel methods, and in particular on structured approximations of distributions of random matrices, this paper proposes Walsh-Hadamard Variational Inference, which uses Walsh-Hadamard-based factorization strategies to reduce model parameterization, accelerate computations, and increase the expressiveness of the approximate posterior beyond fully factorized ones.
Although static networks have been extensively studied in machine learning, data mining, and AI communities for many decades, the study of dynamic networks has recently taken center stage due to the prominence of social media and its effects on the dynamics of social networks. In this paper, we propose a statistical model for dynamically evolving networks, together with a variational inference approach. Our model, Neural Latent Space Model with Variational Inference, encodes edge dependencies across different time snapshots. It represents nodes via latent vectors and uses interaction matrices to model the presence of edges. These matrices can be used to incorporate multiple relations in heterogeneous networks by having a separate matrix for each of the relations. To capture the temporal dynamics, both node vectors and interaction matrices are allowed to evolve with time. Existing network analysis methods use representation learning techniques for modelling networks. These techniques are different for homogeneous and heterogeneous networks because heterogeneous networks can have multiple types of edges and nodes as opposed to a homogeneous network. Unlike these, we propose a unified model for homogeneous and heterogeneous networks in a variational inference framework. Moreover, the learned node latent vectors and interaction matrices may be interpretable and therefore provide insights on the mechanisms behind network evolution. We experimented with a single step and multi-step link forecasting on real-world networks of homogeneous, bipartite, and heterogeneous nature, and demonstrated that our model significantly outperforms existing models.
A recent novel extension of multi-output Gaussian processes handles heterogeneous outputs assuming that each output has its own likelihood function. It uses a vector-valued Gaussian process prior to jointly model all likelihoods' parameters as latent functions drawn from a Gaussian process with a linear model of coregionalisation covariance. By means of an inducing points framework, the model is able to obtain tractable variational bounds amenable to stochastic variational inference. Nonetheless, the strong conditioning between the variational parameters and the hyper-parameters burdens the adaptive gradient optimisation methods used in the original approach. To overcome this issue we borrow ideas from variational optimisation introducing an exploratory distribution over the hyper-parameters, allowing inference together with the posterior's variational parameters through a fully natural gradient optimisation scheme. Furthermore, in this work we introduce an extension of the heterogeneous multi-output model, where its latent functions are drawn from convolution processes. We show that our optimisation scheme can achieve better local optima solutions with higher test performance rates than adaptive gradient methods, this for both the linear model of coregionalisation and the convolution processes model. We also show how to make the convolutional model scalable by means of stochastic variational inference and how to optimise it through a fully natural gradient scheme. We compare the performance of the different methods over toy and real databases.
In many real-world applications of machine learning, data are distributed across many clients and cannot leave the devices they are stored on. Furthermore, each client's data, computational resources and communication constraints may be very different. This setting is known as federated learning, in which privacy is a key concern. Differential privacy is commonly used to provide mathematical privacy guarantees. This work, to the best of our knowledge, is the first to consider federated, differentially private, Bayesian learning. We build on Partitioned Variational Inference (PVI) which was recently developed to support approximate Bayesian inference in the federated setting. We modify the client-side optimisation of PVI to provide an (${\epsilon}$, ${\delta}$)-DP guarantee. We show that it is possible to learn moderately private logistic regression models in the federated setting that achieve similar performance to models trained non-privately on centralised data.
Bayesian neural networks (BNNs) augment deep networks with uncertainty quantification by Bayesian treatment of the network weights. However, such models face the challenge of Bayesian inference in a high-dimensional and usually over-parameterized space. This paper investigates a new line of Bayesian deep learning by performing Bayesian inference on network structure. Instead of building structure from scratch inefficiently, we draw inspirations from neural architecture search to represent the network structure. We then develop an efficient stochastic variational inference approach which unifies the learning of both network structure and weights. Empirically, our method exhibits competitive predictive performance while preserving the benefits of Bayesian principles across challenging scenarios. We also provide convincing experimental justification for our modeling choice.
Approaches to continual learning aim to successfully learn a set of related tasks that arrive in an online manner. Recently, several frameworks have been developed which enable deep learning to be deployed in this learning scenario. A key modelling decision is to what extent the architecture should be shared across tasks. On the one hand, separately modelling each task avoids catastrophic forgetting but it does not support transfer learning and leads to large models. On the other hand, rigidly specifying a shared component and a task-specific part enables task transfer and limits the model size, but it is vulnerable to catastrophic forgetting and restricts the form of task-transfer that can occur. Ideally, the network should adaptively identify which parts of the network to share in a data driven way. Here we introduce such an approach called Continual Learning with Adaptive Weights (CLAW), which is based on probabilistic modelling and variational inference. Experiments show that CLAW achieves state-of-the-art performance on six benchmarks in terms of overall continual learning performance, as measured by classification accuracy, and in terms of addressing catastrophic forgetting.
We present a Gaussian Variational Inference (GVI) technique that can be applied to large-scale nonlinear batch state estimation problems. The main contribution is to show how to fit both the mean and (inverse) covariance of a Gaussian to the posterior efficiently, by exploiting factorization of the joint likelihood of the state and data, as is common in practical problems. This is different than Maximum A Posteriori (MAP) estimation, which seeks the point estimate for the state that maximizes the posterior (i.e., the mode). The proposed Exactly Sparse Gaussian Variational Inference (ESGVI) technique stores the inverse covariance matrix, which is typically very sparse (e.g., block-tridiagonal for classic state estimation). We show that the only blocks of the (dense) covariance matrix that are required during the calculations correspond to the non-zero blocks of the inverse covariance matrix, and further show how to calculate these blocks efficiently in the general GVI problem. ESGVI operates iteratively, and while we can use analytical derivatives at each iteration, Gaussian cubature can be substituted, thereby producing an efficient derivative-free batch formulation. ESGVI simplifies to precisely the Rauch-Tung-Striebel (RTS) smoother in the batch linear estimation case, but goes beyond the 'extended' RTS smoother in the nonlinear case since it finds the best-fit Gaussian (mean and covariance), not the MAP point estimate. We demonstrate the technique on controlled simulation problems and a batch nonlinear Simultaneous Localization and Mapping (SLAM) problem with an experimental dataset.
The reparameterization trick has become one of the most useful tools in the field of variational inference. However, the reparameterization trick is based on the standardization transformation which restricts the scope of application of this method to distributions that have tractable inverse cumulative distribution functions or are expressible as deterministic transformations of such distributions. In this paper, we generalized the reparameterization trick by allowing a general transformation. We discover that the proposed model is a special case of control variate indicating that the proposed model can combine the advantages of CV and generalized reparameterization.
Posterior collapse in Variational Autoencoders (VAEs) arises when the variational posterior distribution closely matches the prior for a subset of latent variables. This paper presents a simple and intuitive explanation for posterior collapse through the analysis of linear VAEs and their direct correspondence with Probabilistic PCA (pPCA). We explain how posterior collapse may occur in pPCA due to local maxima in the log marginal likelihood. Unexpectedly, we prove that the ELBO objective for the linear VAE does not introduce additional spurious local maxima relative to log marginal likelihood. We show further that training a linear VAE with exact variational inference recovers an identifiable global maximum corresponding to the principal component directions. Empirically, we find that our linear analysis is predictive even for high-capacity, non-linear VAEs and helps explain the relationship between the observation noise, local maxima, and posterior collapse in deep Gaussian VAEs.
We conduct non-asymptotic analysis on the mean-field variational inference for approximating posterior distributions in complex Bayesian models that may involve latent variables. We show that the mean-field approximation to the posterior can be well-approximated relative to the Kullback-Leibler divergence discrepancy measure by a normal distribution whose center is the maximum likelihood estimator (MLE). In particular, our results imply that the center of the mean-field approximation matches the MLE up to higher-order terms and there is essentially no loss of efficiency in using it as a point estimator for the parameter in any regular parametric model with latent variables. We also propose a new class of variational weighted likelihood bootstrap (VWLB) methods for quantifying the uncertainty in the mean-field variational inference. The proposed VWLB can be viewed as a new sampling scheme that produces independent samples for approximating the posterior. Comparing with traditional sampling algorithms such Markov Chain Monte Carlo, VWLB can be implemented in parallel and is free of tuning.
A simple and widely adopted approach to extend Gaussian processes (GPs) to multiple outputs is to model each output as a linear combination of a collection of shared, unobserved latent GPs. An issue with this approach is choosing the number of latent processes and their kernels. These choices are typically done manually, which can be time consuming and prone to human biases. We propose Gaussian Process Automatic Latent Process Selection (GP-ALPS), which automatically chooses the latent processes by turning off those that do not meaningfully contribute to explaining the data. We develop a variational inference scheme, assess the quality of the variational posterior by comparing it against the gold standard MCMC, and demonstrate the suitability of GP-ALPS in a set of preliminary experiments.
Stochastic gradient descent (SGD) is the workhorse of modern machine learning. Sometimes, there are many different potential gradient estimators that can be used. When so, choosing the one with the best tradeoff between cost and variance is important. This paper analyzes the convergence rates of SGD as a function of time, rather than iterations. This results in a simple rule to select the estimator that leads to the best optimization convergence guarantee. This choice is the same for different variants of SGD, and with different assumptions about the objective (e.g. convexity or smoothness). Inspired by this principle, we propose a technique to automatically select an estimator when a finite pool of estimators is given. Then, we extend to infinite pools of estimators, where each one is indexed by control variate weights. This is enabled by a reduction to a mixed-integer quadratic program. Empirically, automatically choosing an estimator performs comparably to the best estimator chosen with hindsight.
We develop amortized population Gibbs (APG) samplers, a class of scalable methods that frames structured variational inference as adaptive importance sampling. APG samplers construct high-dimensional proposals by iterating over updates to lower-dimensional blocks of variables. We train each conditional proposal by minimizing the inclusive KL divergence with respect to the conditional posterior. To appropriately account for the size of the input data, we develop a new parameterization in terms of neural sufficient statistics. Experiments show that APG samplers can train highly structured deep generative models in an unsupervised manner, and achieve substantial improvements in inference accuracy relative to standard autoencoding variational methods.